1. If f(x) is an odd degree polynomial with positive leading coefficient, then f(x) →-∞ as x →-∞ and f(x) →∞ as x → ∞.
2. If f(x) is an odd degree polynomial with negative leading coefficient, then f(x) → ∞ as x → -∞ and f(x) →-∞ as x →∞.
3. If the degree of a polynomial f(x) is even and the leading coefficient is positive, then f(x) → ∞ as x → ±∞.
4. If f(x) is an even degree polynomial with negative leading coefficient, then f(x) → -∞ as x →±∞.
Step-by-step explanation:
The Limiting Behavior of Polynomials
The limiting behavior of a function describes what happens to the function as x → ±∞. The degree of a polynomial and the sign of its leading coefficient dictates its limiting behavior. In particular,
If the degree of a polynomial f(x) is even and the leading coefficient is positive, then f(x) → ∞ as x → ±∞.
If f(x) is an even degree polynomial with negative leading coefficient, then f(x) → -∞ as x →±∞.
If f(x) is an odd degree polynomial with positive leading coefficient, then f(x) →-∞ as x →-∞ and f(x) →∞ as x → ∞.
If f(x) is an odd degree polynomial with negative leading coefficient, then f(x) → ∞ as x → -∞ and f(x) →-∞ as x →∞.
Answers & Comments
Answer:
1. If f(x) is an odd degree polynomial with positive leading coefficient, then f(x) →-∞ as x →-∞ and f(x) →∞ as x → ∞.
2. If f(x) is an odd degree polynomial with negative leading coefficient, then f(x) → ∞ as x → -∞ and f(x) →-∞ as x →∞.
3. If the degree of a polynomial f(x) is even and the leading coefficient is positive, then f(x) → ∞ as x → ±∞.
4. If f(x) is an even degree polynomial with negative leading coefficient, then f(x) → -∞ as x →±∞.
Step-by-step explanation:
The Limiting Behavior of Polynomials
The limiting behavior of a function describes what happens to the function as x → ±∞. The degree of a polynomial and the sign of its leading coefficient dictates its limiting behavior. In particular,
If the degree of a polynomial f(x) is even and the leading coefficient is positive, then f(x) → ∞ as x → ±∞.
If f(x) is an even degree polynomial with negative leading coefficient, then f(x) → -∞ as x →±∞.
If f(x) is an odd degree polynomial with positive leading coefficient, then f(x) →-∞ as x →-∞ and f(x) →∞ as x → ∞.
If f(x) is an odd degree polynomial with negative leading coefficient, then f(x) → ∞ as x → -∞ and f(x) →-∞ as x →∞.